CN104950670B - A kind of integrated Multiple Model Control Method of CSTR - Google Patents

Abstract

The invention discloses a kind of integrated Multiple Model Control Method of CSTR, suitable for the control problem of nonlinear system, especially the control problem of first kernel response CSTR, can guarantee that the industrial process quick, accurately, stably from a certain stable operating point can be transformed into another stable operating point.Because the present invention considers influence of the control feedback to system dynamic characteristic, compared with existing first kernel response CSTR Multiple Model Control Methods, advantage is especially notable.

Description

A kind of integrated Multiple Model Control Method of CSTR

Technical field

Divided the present invention relates to a kind of new, space and the integrated multi-model process of optimal control, more particularly to one The Multiple Model Control Method for first kernel response CSTR (CSTR) is planted, belongs to automatic control technology field.

Background technology

CSTR (Continuous Stirred Tank Reactor, CSTR) is a kind of common Nonlinear chemical reaction device, because its low cost, heat-exchange capacity are strong and the features such as product quality stabilization, become production polymerization The nucleus equipment of thing, in dyestuff, pharmaceutical reagent, food and synthetic material industry, is widely used.The control of CSTR Mainly including concentration, temperature etc., the control to these variables will directly influence the quality of chemical products to variable processed.

By the research of scholars' decades, linear control technique has developed quite ripe, but reality industrial process Object is not suitable for being regarded as linear system being controlled system design, such as the process object such as CSTR, rectifying column, and work model Enclose big, set point change is big, and very strong nonlinear characteristic is presented, and traditional linear control theory and technology it is to be difficult to Satisfied control effect.On the other hand, although current nonlinear control techniques have many achievements, but there are still some Limitation, if desired for accurate model, controller design is also very complicated, fails to obtain good application in practice in industry.

One kind compromise, easy and effective solution are multi-model process, and it is based on the plan of " decomposing-synthesis " Slightly, complicated problem is divided into relatively simple minor issue to solve, basic thought is by the operating space of nonlinear system point Into some subregions, a simple Local Linear Model/controller is set up respectively in every sub-regions, finally again by these offices Portion's linear model/controller synthesizes world model/controller, it is intended to using traditional linear control technique nonlinear Control Problem is simplified.

Multi-model process application is relatively broad, also has many scholars to propose the multi-model process framework of uniqueness, in theory Abundant achievement, but relative some ripe control theories are all achieved with practice, multi-model still there are many needs to improve part. Such as, the construction of submodel collection also lacks the theoretical direction of systematization, including the scope of application of each submodel (will also have influence on Synthesis mode, i.e. models switching strategy).At present, typically rule of thumb model quantity and the scope of application are determined in practical application, And most theoretical result then attempts to design some indexs, open cycle system is based on these indexs more, does not often consider closed loop The change that the addition fed back in control brings to system dynamic characteristic.For another example, the scheduling mode between multiple model/controllers It is also a problem, most multi-model process is not the scheduling carried out under Unified frame, and this may not only influence Control effect, in some instances it may even be possible to the unstable of system can be caused.

The content of the invention

The purpose of the present invention is to solve the shortcomings of the prior art, there is provided a kind of integration of CSTR Multiple Model Control Method, the present invention is especially used cold suitable for the control problem of nonlinear system in first kernel response CSTR But the process of liquid temperature control material concentration, enables the system to switch between different stable operating points.

In order to realize above-mentioned purpose, the technical solution used in the present invention is as follows：A kind of CSTR Integrated Multiple Model Control Method, the method is that based on Hybrid System Theory, space divides and optimal control integration For the Multiple Model Control Method of first kernel response CSTR, comprise the following steps：

(1) for the irreversible exothermic reaction of single order, state-space model is built, as shown in Equation 1：

Wherein, state x₁It is the concentration C of reactant_A, x₁∈[0,1]；x₂It is temperature in the kettle T, x₂∈[0,6]；Y is that output becomes Amount, y ∈ [0,1]；U is coolant temperature T_c0, u ∈ [- 2,2]；The concentration C of reactant_ARate of change,It is temperature in the kettle The rate of change of T, γ=20, B=8, D_a=0.072, β=0.3.Above-mentioned all variables are all dimensionless.

U=0 is taken, three stable operating point x of the process are obtained_e1、x_e2、x_e3, respectively：

x_e1=[x₁,x₂]^T=[0.8560,0.8859]^T；

x_e2=[x₁,x₂]^T=[0.5528,2.7517]^T；

x_e3=[x₁,x₂]^T=[0.2353,4.7050]^T。

(2) nonlinear model in step 1 is linearized near each operating point for obtaining, is obtained three parts Linear model, and transcription is hybrid model；Specially：

(2.1) nonlinear model is linearized at operating point, is obtained piecewise linearity affine model as follows：

In formula, M is operating point quantity；A_m、B_m、C_m、D_mFor the nonlinear model differential equation in each operating point to state x's First-order partial derivative determinant, or Jacobian matrix；And a_m=-A_mx_em-B_mu_em, c_m=y_em-C_mx_em-D_mu_em；Ω_mIt is each part The scope of application of model, x ∈ Ω_mI.e. representation space is divided.

Three stable operating point x that step 1 is obtained_e1、x_e2、x_e3Formula 2 is updated to, three Local Linear Models is obtained such as Under：

(2.2) space divides x ∈ Ω_mCan be simplified with inequality group and represented, as shown in Equation 4：

x∈Ω_m→E_mx<F_m(4)

In formula, E_m, F_mIt is the appropriate constant matrices of dimension, waits to ask.

Hybrid system mode is incorporated into above-mentioned formula 2, hybrid model is obtained：

In formula, i represents the mode of hybrid system, and k represents mode sequence order, is nonnegative integer, i_kValue in from 1 to M, Therefore i (t) is step function, τ_kIt is k-th initial time of period.

(3) according to hybrid model, design performance index sets up hybrid system optimal control problem, using optimality Condition solves value function, and determines that space divides；Specially：

(3.1) hybrid model obtained according to step 2, sets up optimal control problem, and performance indications are as follows：

L represents operating cost in formula, and performance indications J is in integrated form.

(3.2) Dynamic Programming is used, optimal modal sequence i and optimum control input u is found according to formula 6, make J minimum, from And obtain corresponding value function：

I in formula₀It is initial mode, x is original state, t₀It is initial time.

(3.3) according to formula 5 and formula 7, further obtained by HJB (Hamilton-Jacobi-Bellman) equation solution Value function V, the HJB equations are：

V in formula_xIt is gradients of the value function V on x.

(3.4) the value function V obtained according to step 3.3, obtains the point set at Mode-switch

(3.5) E is divided according to space_mx<F_mAnd the point set that step 3.4 is obtainedDetermine by least square method E_m、F_mValue.

(4) after obtaining piecewise linearity affine model and space division, piecewise linearity affine model is converted into mixed logic Dynamic (MixedLogical Dynamical, MLD) model, designs MPC controller under this Unified frame, solves MIP problems Control signal is obtained, implements control；Specially：

(4.1) according to E_m、F_mValue, the piecewise linearity affine model that step 2 is obtained is converted to MLD models, described MLD model criteria forms are：

(4.2) performance indications described in linear restriction according to formula 10 and formula 6, design MPC controller solves following problem：

In formula, N is prediction duration, and k is current time, and x (k) is current time state value, normal form and parameter and property Energy index is corresponding,Be optimum control list entries, i.e. MPC problems Solution.

If optimal solutionIn the presence of, according to the thought of MPC rolling optimizations, an input that first was worth as the k moment, i.e., U (k)=u^*(k)。

U (k) is converted into industry standard signal by D/A converter, transmitter, is applied in the regulating valve of hot water pipeline To implement control.

(5) in subsequent time, i.e. k+1, step 4 is repeated.

(6) if state value x (k) at current time is unknowable, by designing the EFK of self adaptation, work as controller is provided The estimate of preceding unknown state, specially：

By the system that the piecewise linearity affine model of formula 2 is discrete, identical with the MLD in step 4.1 using the time and virtual Noise w (k) represents the model mismatch existed between piecewise linearity affine model and real system, obtains estimator and uses Model it is as shown in Equation 12：

W (k) is virtual system noise in formula, is that average is q (k), the Gaussian noise that covariance is Q (k), and q (k), Q K () is unknown；A_diObtain Deng the model by formula 2 is discrete.

According to formula 12, in each moment k, inputoutput data and current outputting measurement value according to history can design EFK Prediction, renewal equation and suboptimum unbiased maximum posteriori Noise statistics extimators estimate noise statisticses and state to replace, and by shape State estimate is input to MPC controller.

The beneficial effects of the invention are as follows the method for the present invention ensure that first kernel response CSTR can quickly, accurately, stably Another stable operating point is transformed into from a certain stable operating point.Advantages of the present invention is essentially consisted in and just known that after submodel collection also Can implement, can be used for most of non-linear objects, and take into account influence of the control feedback to system dynamic characteristic, compared to preceding Design objective in people's achievement has more theoretical property, wide usage, and can also process state the situation common in practice such as can not survey.This Invention is compared with existing multi-model process, and advantage is especially notable.

Brief description of the drawings

Fig. 1 is the application scenarios schematic diagram of CSTR production processes of the present invention.Wherein, C_A0For substance A input concentration, T₀It is feeding temperature, q feed rates, C_AConcentration, T when being chemically reacted for substance A are temperature of reactor, q_cIt is coolant Flow, T_c0It is coolant temperature.

Fig. 2 is the closed loop optimum state space division result tried to achieve based on hybrid system optimal control problem in the present invention Schematic diagram.

Fig. 3 is whole control system architecture schematic diagram.

Fig. 4 is that multi-model process and the method for the invention based on gap metric is respectively adopted, to first kernel response CSTR processes, the simulation result figure that operating point 1 is controlled is switched to from operating point 3, and in figure, Y* is patent description side of the present invention Method curve obtained, Y_sIt is then former approach curve obtained.

Specific embodiment

In order to improve the shortcoming of conventional multi-mode type method, the present invention proposes a kind of space and divides based on Hybrid System Theory With the multi-model process of optimal control integration, this process object of CSTR is applicable not only to, can also be applied to general non-thread Sexual system control problem.To become apparent from the object, technical solutions and advantages of the present invention, this method is used for single order below anti- Answer CSTR operating points handoff procedure.Specific implementation step is as follows：

1st, carry out state space modeling to first kernel response CSTR process objects, and determine its normal work working range, The information such as input constraint, steady operation point.

The CSTR processes considered in the present invention, have been carried out much by many scholars as classical non-linear object Research.As shown in figure 1, what is carried out in kettle is the irreversible exothermic reaction of single order, i.e., toward addition something A in CSTR, there is chemistry Reaction product matter B, relatively more representational example such as styrene bulk thermal polymerization process.According to forefathers' achievement, one can be set up The state-space model of rank reaction CSTR is as follows：

Wherein, state x₁It is the concentration C of reactant A_A, x₂It is temperature in the kettle T；Y is output variable, is also C_A；U is control Variable, refers to coolant temperature T_c0.Here all variables are all nondimensional, and equation preset parameter is D_a=0.072, γ=20, B =8, and β=0.3, and the span of variable is x₁∈[0,1],x₂∈[0,6],u∈[-2,2],y∈[0,6]。

There are three stable operating points in the process, operating point x is tried to achieve by taking u=0_e1、x_e2、x_e3It is as follows：

x_e1=[x₁,x₂]^T=[0.8560,0.8859]^T；

x_e2=[x₁,x₂]^T=[0.5528,2.7517]^T；

x_e3=[x₁,x₂]^T=[0.2353,4.7050]^T。

2nd, the nonlinear model is linearized near each operating point, is obtained scope of application submodel undetermined Collection, and transcription is hybrid system form.

Nonlinear model is linearized at each operating point, piecewise linear model form is can obtain as follows：

In formula, M is operating point quantity；A_m、B_m、C_m、D_mFor the nonlinear model differential equation in each operating point to state x's First-order partial derivative determinant, or Jacobian matrix；And a_m=-A_mx_em-B_mu_em, c_m=y_em-C_mx_em-D_mu_em；Ω_mIt is each part The scope of application of model, x ∈ Ω_mI.e. representation space is divided.

3 Local Linear Models will be can obtain after the state-space model linearisation of CSTR：

The now scope of application Ω of each partial model_mIt is undetermined, asked to simplify space partition problem and follow-up controller Solution problem, it is assumed that boundary condition can represent with inequality group, i.e.,

x∈Ω_m→E_mx<F_m (4)

In formula, E_m, F_mIt is the appropriate constant matrices of dimension, value is undetermined.

The concept of hybrid system mode is introduced, makes i for mode sequence, above-mentioned equation can be rewritten as：

In formula, i represents the mode of hybrid system, and k represents mode sequence order, is nonnegative integer, i_kValue in from 1 to M, Therefore i (t) is step function, τ_kIt is k-th initial time of period.

3rd, according to hybrid model, design performance index sets up hybrid system optimal control problem, using optimality bar Part solves value function, and determines that closed loop optimum state space divides.

In order to determine E_m, F_mValue, obtain consider control feedback closed loop optimum state space divide, need to be according to equation (5) set up optimal control problem and solve.The performance indications of usual hybrid system optimal control problem are by operating cost and switching Cost two is constituted, and wherein switching surfaces refer to the resource consumption that actual hybrid system Mode-switch causes, and can suppress mode frequently Switch numerously.Only it is to be converted to PWA to mix to be but nonlinear system involved in the present invention has no mix property in itself System, mode switches and has no materially affect, therefore the present invention does not consider switching surfaces, and performance indications form is as follows：

L represents operating cost in formula, and performance indications J is in integrated form.

Now need to find optimal modal sequence i and optimum control is input into u and makes J minimum.Solve hybrid system optimal control problem One of method be Dynamic Programming.Corresponding value function can be designed by equation (6)：

I in formula₀It is initial mode, x is original state, t₀It is initial time.

Then further there is HJB (Hamilton-Jacobi-Bellman) equation：

V in formula_xIt is gradients of the value function V on x.The algorithm non-invention emphasis of numerical solution HJB equations, does not open up herein Open explanation.After value function V is asked for, the point set at Mode-switch can be obtained with following formula：

In correlative study and practice, the performance indications of quadratic form are relatively conventional, therefore for first kernel response CSTR processes, this hair Bright design performance index form is as follows：

Y in formula_e, u_eIt is setting value, Q, R nonnegative definite matrix represent weights.

It is setting value, i.e. y to make operating point 1_e=0.8560, u_e=0, then Q=500, R=1 are taken, corresponding value function is solved, After obtaining the point set on Mode-switch border, then E is can determine that by least square method_i、F_iValue, final closed loop is optimal State space division result can be found in Fig. 2.

(4) after obtaining submodel collection and optimal boundary, MLD models are converted to, MPC controls is designed under this Unified frame Device, solves MIP problems and obtains control signal, implements control.

This step is related to the design of Multiple model control device.The synthesis mode of common direct-cut operation and soft handover is different from, this Invention is used and carries out global optimization control based on MLD modellings MPC controller.MLD models are used for the physics by interacting The modeling of the system described by rule, logic rules and operation constraint, by by about in the linear of linear mixed-integer inequality Dynamical equation group is represented.

Submodel collection and optimal boundary have been obtained, need to (sampling time can be selected according to actual conditions by PWA models discretization Take, the emulation in accompanying drawing takes T=0.2), such as following form：

Introduce logical variable treatment boundary condition：

Introduce auxiliary variable：

It is ultimately converted to the MLD models of standard：

To divide space and optimal control integration, then performance according to the linear restriction of equation (14) and hereinbefore is needed Index, design MPC controller solves following problem：

In formula, N is prediction duration, and k is current time, and x (k) is current time state value, normal form and parameter and property Energy index thinks correspondence,It is that optimum control list entries, i.e. MPC are asked The solution of topic.

This MPC Optimization Solutions problem is asked to solve MIP problems using existing instrument such as ILOG CPLEX softwares Solution.If optimal solutionIn the presence of according to the thought of MPC rolling optimizations, the input that only first was worth as the k moment, i.e. u (k) =u^*(k).In subsequent time, i.e. k+1, Optimization Solution step will be repeated.

Control signal u (k) at k moment is exported by D/A converter, then industry standard signal (4 is converted into through transmitter ~20mA), apply in the regulating valve of hot water pipeline in fig. 1, adjust hot water flow to change coolant temperature, so as to control The concentration of substance A.

(5) if consideration state can not survey or measure time lag greatly and there is the situation of system noise, self adaptation can be designed EFK, the estimate of current unknown state is provided for controller.

It is not difficult to find out, the multi-model process of present invention design need to obtain the state value at current time, be solved for controller. But in the industrial process of reality, the immesurable situation of some states is often there is, or measure time lag is big, noise is big etc., influence The control effect of MPC.Therefore design point estimator is needed, in known current output and history inputoutput data, provide current shape The estimate of state.

Consideration is output as actual value, introduces virtual noise in discrete piecewise linearity reflection model to represent segmented line The model mismatch factor existed between sexual reflex model and real system, can obtain the model that estimator uses as follows：

W (k) is virtual system noise in formula, is that average is q (k), the Gaussian noise that covariance is Q (k), and q (k), Q K () is unknown.

First assume q (k), Q (k), it is known that there are EFK accounting equations as follows：

In formulaThe as estimate of current time state.

Design suboptimum unbiased maximum posteriori Noise statistics extimators estimate q (k), Q (k), and recurrence equation is as follows：

After setting initial value, noise statisticses and state can be alternately estimated by equation (17), (18), and state estimation is defeated Enter to MPC controller (control structure is referring to Fig. 3).

Claims (1)

1. the integrated Multiple Model Control Method of a kind of CSTR, it is characterised in that, the method includes following Step:

(1) for the irreversible exothermic reaction of single order, state-space model is built, as shown in Equation 1：

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=-x_1+D_a\&CenterDot;(1-x_1)\&CenterDot;\exp \left(\frac{x_2}{1+x_2/\mathrm{\&gamma;}}\right)\\ {\stackrel{\&CenterDot;}{x}}_2=-x_2+B\&CenterDot;D_a\&CenterDot;(1-x_1)\&CenterDot;\exp \left(\frac{x_2}{1+x_2/\mathrm{\&gamma;}}\right)+\mathrm{\&beta;}\&CenterDot;(u-x_2)\\ y=x_1\end{array}\right.---\left(1\right)$

Wherein, state x₁It is the concentration C of reactant_A, x₁∈[0,1]；x₂It is temperature in the kettle T, x₂∈[0,6]；Y is output variable, y ∈[0,1]；U is coolant temperature T_c0, u ∈ [- 2,2]；The concentration C of reactant_ARate of change,It is temperature in the kettle T's Rate of change, γ=20, B=8, D_a=0.072, β=0.3；Above-mentioned all variables are all dimensionless；

U=0 is taken, three stable operating point x of the process are obtained_e1、x_e2、x_e3, respectively：

x_e1=[x₁,x₂]^T=[0.8560,0.8859]^T；

x_e2=[x₁,x₂]^T=[0.5528,2.7517]^T；

x_e3=[x₁,x₂]^T=[0.2353,4.7050]^T；

(2) nonlinear model in step 1 is linearized near each operating point for obtaining, is obtained three local linears Model, and transcription is hybrid model；Specially：

(2.1) nonlinear model is linearized at operating point, is obtained piecewise linearity affine model as follows：

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}=A_{m}x+B_{m}u+a_m,\\ y=C_{m}x+D_{m}u+c_m,\end{array}\right.,x\&Element;{\mathrm{\&Omega;}}_m,m=1,...,M---\left(2\right)$

In formula, M is operating point quantity；A_m、B_m、C_m、D_mFor the nonlinear model differential equation is inclined to the single order of state x in each operating point Jacobian determinant, or Jacobian matrix；And a_m=-A_mx_em-B_mu_em, c_m=y_em-C_mx_em-D_mu_em；Ω_mIt is each partial model The scope of application, x ∈ Ω_mI.e. representation space is divided；

Three stable operating point x that step 1 is obtained_e1、x_e2、x_e3Formula 2 is updated to, three Local Linear Models can be obtained such as Under：

$\begin{array}{c}\stackrel{\&CenterDot;}{x}=\left[\begin{array}{cc}-1.1682& -0.1320\\ 1.3455& -0.2439\end{array}\right]x+\left[\begin{array}{c}0\\ 0.3\end{array}\right]u+\left[\begin{array}{c}1.1165\\ -0.9351\end{array}\right],y=x_1\\ \stackrel{\&CenterDot;}{x}=\left[\begin{array}{cc}-1.8088& -0.3455\\ 6.4705& 1.4640\end{array}\right]x+\left[\begin{array}{c}0\\ 0.3\end{array}\right]u+\left[\begin{array}{c}1.9504\\ -7.6047\end{array}\right],y=x_1\\ \stackrel{\&CenterDot;}{x}=\left[\begin{array}{cc}-4.2474& -0.5008\\ 25.9792& 2.7063\end{array}\right]x+\left[\begin{array}{c}0\\ 0.3\end{array}\right]u+\left[\begin{array}{c}3.3557\\ -18.8460\end{array}\right],y=x_1\end{array}---\left(3\right)$

(2.2) space divides x ∈ Ω_mCan be simplified with inequality group and represented, as shown in Equation 4：

x∈Ω_m→E_mx<F_m (4)

In formula, E_m, F_mIt is the appropriate constant matrices of dimension, waits to ask；

Hybrid system mode is incorporated into above-mentioned formula 2, hybrid model is obtained：

$\left\{\begin{array}{cc}\stackrel{\&CenterDot;}{x}=A_{i}x+B_{i}u+a_i,& \\ y=C_{i}x+D_{i}u+c_i,& i_k,i_{k+1}\&Element;I=\{1,2,\mathrm{..},M\}\\ i\left(t\right)=i_k,t\&Element;\&lsqb;{\mathrm{\&tau;}}_k,{\mathrm{\&tau;}}_{k+1}),& k=0,1,...\\ {\mathrm{\&tau;}}_{k+1}=\inf \left\{t\right|E_{i_k{i}_{k+1}}x\left(t\right)<F_{i_k{i}_{k+1}}\}& \end{array}\right.---\left(5\right)$

In formula, i represents the mode of hybrid system, and k represents mode sequence order, is nonnegative integer, i_kValue in from 1 to M, therefore i T () is step function, τ_kIt is k-th initial time of period；

(3) according to hybrid model, design performance index sets up hybrid system optimal control problem, using optimality condition Value function is solved, and determines that space divides；Specially：

(3.1) hybrid model obtained according to step 2, sets up optimal control problem, and performance indications are as follows：

$J(i,u)={\&Integral;}_0^{\mathrm{\&infin;}}L\left(x\right(t),u(t),y(t\left)\right)dt---\left(6\right)$

L represents operating cost in formula, and performance indications J is in integrated form；

(3.2) Dynamic Programming is used, optimal modal sequence i and optimum control input u is found according to formula 6, make J minimum, so that To corresponding value function：

$V(i_0,x)=\underset{u}{inf}\&lsqb;{\&Integral;}_{t_0}^{\mathrm{\&infin;}}L\left(x\right(s),u(s),y(s\left)\right)ds\&rsqb;---\left(7\right)$

I in formula₀It is initial mode, x is original state, t₀It is initial time；

(3.3) according to formula 5 and formula 7, value letter is further obtained by HJB (Hamilton-Jacobi-Bellman) equation solution V is counted, the HJB equations are：

$\begin{array}{c}\min \{\underset{u}{\min }(A_{i_0}x+B_{i_0}u+a_{i_0})V_x(i_0,x)+L(x,y,u),\\ \underset{j\&NotEqual;i_0}{\min }\{V(j,x)-V(i_0,x)\}\}=0\end{array}---\left(8\right)$

V in formula_xIt is gradients of the value function V on x；

(3.4) the value function V obtained according to step 3.3, obtains the point set at Mode-switch

$X_{i_{0}j}^{*}=\left\{x\right|V(j,x)-V(i_0,x)=0\}---\left(9\right)$

(3.5) E is divided according to space_mx<F_mAnd the point set that step 3.4 is obtainedDetermine E by least square method_m、F_m Value；

(4) after obtaining piecewise linearity affine model and space division, piecewise linearity affine model is converted into integrating mixed logic dynamic (MixedLogical Dynamical, MLD) model, designs MPC controller under this Unified frame, solves MIP problems and obtains Control signal, implements control；Specially：

(4.1) according to E_m、F_mValue, the piecewise linearity affine model that step 2 is obtained is converted to MLD models, the MLD moulds Type canonical form is：

$\left\{\begin{array}{c}x(k+1)=Ax\left(k\right)+B_{1}u\left(k\right)+B_2\mathrm{\&delta;}\left(k\right)+B_{3}z\left(k\right)\\ y\left(k\right)=Cx\left(k\right)+D_{1}u\left(k\right)+D_2\mathrm{\&delta;}\left(k\right)+D_{3}z\left(k\right)\\ E_2\mathrm{\&delta;}\left(k\right)+E_{3}z\left(k\right)\&le;E_{1}u\left(k\right)+E_{4}x\left(k\right)+E_5\end{array}\right.---\left(10\right)$

(4.2) performance indications described in linear restriction according to formula 10 and formula 6, design MPC controller solves following problem：

$\begin{array}{c}\underset{\left\{u_k^{N-1}\right\}}{min}J(u_k^{N-1},x(k\left)\right)=\underset{i=0}{\overset{N-1}{\&Sigma;}}\left(\right|\left|u\right(k+i)-u_e|{|}_{Q_1}^P\\ +\left|\right|x\left(i\right|k)-x_e|{|}_{Q_2}^P+||y\left(i\right|k)-y_e|{|}_{Q_3}^P)\end{array}---\left(11\right)$

In formula, N is prediction duration, and k is current time, and x (k) is current time state value, and normal form and parameter refer to performance Mark is corresponding,Be optimum control list entries, i.e. MPC problems Solution；

If optimal solutionIn the presence of according to the thought of MPC rolling optimizations, the input that only first was worth as the k moment, i.e. u (k) =u^*(k)；

U (k) is converted into industry standard signal by D/A converter, transmitter, is applied in the regulating valve of hot water pipeline real Apply control；

(5) in subsequent time, i.e. k+1, step 4 is repeated；

(6) if state value x (k) at current time is unknowable, by designing the EFK of self adaptation, for controller is provided currently not Know the estimate of state, specially：

By the system noise that the piecewise linearity affine model of formula 2 is discrete, identical with the MLD in step 4.1 using the time and virtual W (k) represents the model mismatch existed between piecewise linearity affine model and real system, obtains the mould that estimator is used Type is as shown in Equation 12：

$\begin{array}{c}\left\{\begin{array}{c}x\left(k\right)=A_{{\mathrm{di}}_{k-1}}(k-1)+B_{{\mathrm{di}}_{k-1}}u(k-1)+a_{{\mathrm{di}}_{k-1}}+w(k-1)\\ y\left(k\right)=C_{{\mathrm{di}}_k}x\left(k\right)+D_{{\mathrm{di}}_k}u\left(k\right)+c_{{\mathrm{di}}_k}\end{array}\right.\\ E_{i_k}x\left(k\right)<F_{i_k},i_k\&Element;\{1,\mathrm{...},M\}\end{array}---\left(12\right)$

W (k) is virtual system noise in formula, is that average is q (k), the Gaussian noise that covariance is Q (k), and q (k), Q (k) It is unknown；A_diObtain Deng the model by formula 2 is discrete；

According to formula 12, in each moment k, inputoutput data and current outputting measurement value according to history can design EFK pre- Survey, renewal equation and suboptimum unbiased maximum posteriori Noise statistics extimators estimate noise statisticses and state to replace, and by state Estimate is input to MPC controller.